as in (REF ), then one can immediately bound \(\tau \) , the time required for all z-scores to cross a threshold that scales with \(\sqrt{2\log (1/c)}\) , by a term of order \(\Gamma _{\theta }\log (1/c)\) . Almost sure convergence of \(p_{t,i}\)  ensures (REF ) is eventually accurate for each sample path. As observed in [1]}, however, to bound the expected value of a stopping time, we need to make sure the expected time elapsed before the approximation (REF ) is accurate is itself finite. For a different top-two sampling algorithm, [1]} does this by showing a slightly stronger notion of convergence: for each \(\epsilon >0\)  there is a (sample path dependent) time \(T\)  with \(|p_{t,i} - p^*_i|\leqslant \epsilon \)  for \(t\geqslant T\)  and moreover \(\mathbb {E}[T]<\infty \) . [3]} extends this argument to apply to top-two Thompson sampling, i.e. to DTS in the setting without contexts.
